Page 97 - Robotics Designing the Mechanisms for Automated Machinery

86 Dynamic Analysis of Drives

and for the particular solution we have

Substituting these solutions in the homogeneous form of Equation (3.83) and in its
complete form, respectively, we obtain

Using the initial conditions that for t = 0 the speed co = 0, we obtain for the constant A

Thus, the complete solution has the form

The next step is to calculate the 0(Z) dependence. This can obviously be achieved by
direct integration of solution (3.88):

or

For the second assumption, we introduce into the excitation torque a "saw"-like
periodic component. To do so we must express this "saw" in a convenient form, i.e.,
describe it in terms of a Fourier series. Let us approximate this "saw" by inclined straight
lines, as shown in Figure 3.21 (the reader can make another choice for the approxi-
mation form). Then, this periodic torque component T p can be described analytically
by the expression

FIGURE 3.21 Approximation of the
"saw"-like characteristic (see Figure
3.18) of a stepper motor by inclined
straight lines.